A note on the scaling limits of random P\'olya trees

TL;DR

This paper refines the understanding of the structure of random Pólya trees, showing that attached forests are of size proportional to log n, and provides combinatorial and probabilistic insights into their formation.

Contribution

It improves bounds on forest sizes in Pólya trees using analytic combinatorics and offers a new combinatorial interpretation of their weights and structure.

Findings

Attached forests are of size Θ(log n) in random Pólya trees.

Provides a combinatorial interpretation of forest weights via automorphisms.

Derives the probability distribution for the size of forests attached to nodes.

Abstract

Panagiotou and Stufler (arXiv:1502.07180v2) recently proved one important fact on their way to establish the scaling limits of random P\'{o}lya trees: a uniform random P\'{o}lya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements on the bound of $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given P\'{o}lya tree. Finally, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size.